Theory

Collective Variables

In most cases, it is impossible to extract clear information about a system of interest by monitoring the Cartesian coordinates of all atoms directly, especially if our system contains many atoms. Instead, it is possible to monitor the system by defining functions of those coordinates that describe the chemical properties that we are interested in. These functions are called Collective Variables (CVs) and allow biasing specific degrees of freedom or analyzing how those properties evolve. Plumed has numerous CVs already implemented that can be used with ASE. For a complete explanation of CVs implemented in Plumed, go to this link.

Metadynamics

Metadynamics is an enhanced sampling method that allows exploring the configuration landscape by adding cumulative bias in terms of some CVs. This bias is added each $\tau$ time lapse and usually its shape is Gaussian. In time t, the accumulated bias is defined as

(1)

\[V_{B}({\mathbf{s}}, t) = \sum_{t'=\tau, 2\tau,...}^{t' < t}W(\mathbf{s}, t') \hspace{0.1cm} \exp\left({-\sum_i\frac{[s_i\hspace{0.1cm} - \hspace{0.1cm} s_i(t')]^2}{2\sigma_i}}\right) ~,\]

where $\mathbf{s}$ is a set of collective variables, $\sigma_i$ is the width of the Gaussian related to the i-th collective variable, and W(s, t’) is the height of the Gaussian in time t’. In simple metadynamics, W(s, t’) is a constant, but in Well-Tempered Metadynamics, the height of the Gaussians is lower where previous bias was added. This reduction in the height of the new Gaussians decreases the error and avoids exploration towards high free energy states that are thermodynamically irrelevant. The height in time t’ for Well-Tempered Metadynamics is

(2)

\[W(\mathbf{s}, t') = W \exp\left({-\frac{\beta \hspace{0.1cm} V_B({\bf s}, \hspace{0.1cm}t')}{\gamma}}\right) ~,\]

with $W$ the maximum height of the Gaussians, $\beta$ the inverse of the thermal energy ($1/k_BT$) and $\gamma$ a bias factor greater than one that regulates how fast the height of the bias decreases: the higher the bias factor, the slower the heights decrease. Note that when $\gamma$ approaches infinity, this equation becomes constant and simple metadynamics is recovered. In contrast, when $\gamma$ approaches zero, no bias is added, which is the case of Molecular Dynamics.

The addition of the bias potential produces an extra force in each atom, such that the resultant force for the i-th atom is

(3)

\[{\bf F^B}_i = {\bf F}_i - \frac{\partial {\bf s}}{\partial {\bf R}_i} \frac{\partial V_B({\bf s}, t)}{\partial {\bf s}} ~,\]

where $F_i$ is the natural unbiased force, $R_i$ is the coordinate of the atom, and the second term is the additional force due to the added bias.

Part of the power of metadynamics is that it can be used for exploring conformations. Moreover, the accumulated bias converges to the free energy surface ( $F(s)$ ),

(4)

\[\lim_{t\rightarrow \infty} V_B ({\bf{s}}, t) = -\frac{(\gamma -1)} {\gamma} F({\bf s})~.\]
← Introduction
Toy model: Planar 7-Atoms Cluster →